Title: CLTs for Pair Dependent Statistics of Circular Beta Ensembles
Speaker: Ander Aguirre (Ohio State University)
Abstract: In this talk we give an overview of recent results on the fluctuation of the statistic \sum_{i\neq j} f(L_N(\theta_i-\theta_j)) for the Circular Beta Ensemble in the global, mesoscopic and local regimes. This work is morally related to Johansson's 1988 CLT for the linear statistic \sum_i f(\theta_i) and Lambert's subsequent 2019 extension to the mesoscopic regime. The special case of the CUE (\beta=2) in the local regime L_N=N is motivated by Montgomery's study of pair correlations of the rescaled zeros of the Riemann zeta function. Our techniques are of combinatorial nature for the CUE and analytical for \beta\neq2 .